Representation of j=1 rotation matrix

[jdstokes]

[SOLVED] Representation of j=1 rotation matrix

The derivation of this involves the use of the following fact for j=1:

[atex]\frac{J_y}{\hbar} = (J_y/\hbar)^3[/itex].

Is there a simple way to see this other than slogging through the algebra by expanding out the RHS using $J_y = \frac{1}{2i}(J_+ - J_i)$ and $J_{\pm}|jm\rangle = \hbar\sqrt{(j\mp m)(j \pm m + 1)}| j,m\pm 1\rangle$?

 

[pam]

The derivation of this involves the use of the following fact for j=1:
[atex]\frac{J_y}{\hbar} = (J_y/\hbar)^3[/itex].

That's not true in general, since the eigenvalues of J_y for j=1 are +1,0,-1.
It is not true for j=-1. Perhaps there is a special circumstance in your problem.

 

[jdstokes]

Hi Pam,

It is true for j=1, and I was being stupid anyway, the trick is to use matrix multiplication.

Ie just write down $\langle j'=1,m'| J_y|j=1,m\rangle$ and do the trivial matrix multiplication. Doh!

 

[pam]

Of course, I made the silly mistake of thinking (-1)^3=+1.
Using the evs is a simpler proof than even trivial matrix math.

 

[jdstokes]

evs?